A question on Molecular Orbital theory

[tomothy]

My question is as follows. When (qualitatively at least) constructing molecular orbitals for a given molecule, a few "rules" are used to construct them. I'm wondering why the "rules" exist at all.

For example, it is given that orbitals must have "similar energies" (however one quantifies "similar") for them to interact. What is the justification for this? Why is it that in diatomic nitrogen the 2σ orbital is raised in energy above the π orbitals? I've been told it's due to s-p mixing because for nitrogen (compared to oxygen and fluorine) the energy gap between the 2s and 2p orbitals is smaller, thus the orbitals interact to a certain degree. But why is it the orbitals must be of similar energy to interact in the first place. I'd similarly like to know why it is that the HOMO-LUMO energy gap affects how fast reactions, such as nucleophilic addition the carbonyls, occur.

Related to this, I'd like to know why orbitals must have similar orientation to interact. I understand that if for example a 2p_z and 2s orbital were to overlap on two different atoms, it would yield nothing due to a mixture of constructive and destructive overlap, but surely these are possible linear combinations of orbitals thus must still "happen" even if they only exist as virtual states.

Thank you all in advance

 

[niallj]

The degree of interaction between two orbitals is related to how much they overlap with each other. Two orbitals which are of similar energies overlap much more than two of very different energies.

I don't remeber much about rates, I'm afraid, but I assume that if you have a very large HOMO-LUMO gap in the electrophile, then the electrons are going to be added to it at a higher energy than if the HOMO-LUMO gap were smaller.

The orientation point goes back to the overlap. You're right- you can construct any combination of orbitals that you like. If the overlap is very small, however, you basically just get the same two orbitals that you put in.

 

[DrDu]

Being of similar size is one condition for a strong bond between two orbitals. Independently of that, they should be of similar energy. Compare it to two coupled pendulums. If the frequency of the pendulums is equal, energy will oscillate hence and forth between the two and their joint frequency for in phase motion will be considerably lower than that of a single pendulum. If the two pendulums have different frequencies, then the oscillation will mainly be restricted to either one of them.

More directly, the interaction of two orbitals is obtained by diagonalizing a two by two matrix whose diagonal entries are the energies of the isolated orbitals, say E_a and E_b and an outer diagonal interaction V, which depends on the overlap of the two orbitals. The eigenvalues of that matrix are then $((E_b+E_a)\pm \sqrt{(E_b-E_a)^2+4V^2})/2$. If E_a equals E_b, then the bonding energy is V. On the other hand if V is much smaller than the energy difference, then, if E_a is the lower of the two orbitals, its energy will only be lowered by V^2/(E_b-E_a) which is much smaller than V.